22E Page 52
the sum of 2 psi streams. This problem is dealt with in 22W(a).
22E THE SUM OF TWO STREAMS
(a) The Proportional Bulge.
In calculating the frequency of various letters or groups of letters in the sum of two streams whose letter frequencies are known, it is sometimes more convenient to consider the proportional bulges of the letters concerned and not their frequencies. The PB has been introduced in 21(j) and is normally denoted by a small Greek letter.
Consider a stream of letters (U) drawn from an alphabet of r letters,
|Summing over the r letters of the alphabet we get|
(b) The Faltung theorem (a special form of a result stated in 21(m)).
In a stream of letters (U + V) which is the sum of two streams U and V it is clear that
|If every = 0, then = 0||(E3)|
Therefore if two streams, one of which is random, are added together the resulting stream is random.
(c) Multiplication of PB's.
If we put r = 2 and consider the sum of two streams each consisting of dots and crosses, we get
|But ζ. + ζx = 0|